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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMar 31st 2019
   • (edited Mar 31st 2019)
   • PermaLink
   Author: Urs
   Format: MarkdownItexJust for completeness, to go with the entries for the other [[rotation groups]] in [[low-dimensional topology|low]] [[dimensions]]: | [[special orthogonal group]] | [[spin group]] | [[pin group]] | |------------------------------|----------------|---------------| | [[SO(2)]] | [[Spin(2)]] | [[Pin(2)]] | | [[SO(3)]] | [[Spin(3)]] | | | [[SO(4)]] | [[Spin(4)]] | | | | [[Spin(5)]] | | | | [[Spin(6)]] | | | | [[Spin(7)]] | | | [[SO(8)]] | [[Spin(8)]] | | | [[SO(9)]] | [[Spin(9)]] | | <a href="https://ncatlab.org/nlab/revision/Spin%289%29/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Spin%289%29">current</a>

   Just for completeness, to go with the entries for the other rotation groups in low dimensions:

   special orthogonal group	spin group	pin group
   SO(2)	Spin(2)	Pin(2)
   SO(3)	Spin(3)
   SO(4)	Spin(4)
   	Spin(5)
   	Spin(6)
   	Spin(7)
   SO(8)	Spin(8)
   SO(9)	Spin(9)

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeMay 17th 2019
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexAdded a more recent reference * Maurizio Parton, Paolo Piccinni, _The Role of Spin(9) in Octonionic Geometry_, ([arXiv:1810.06288](https://arxiv.org/abs/1810.06288)) <a href="https://ncatlab.org/nlab/revision/diff/Spin%289%29/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/Spin%289%29/2">v2</a>, <a href="https://ncatlab.org/nlab/show/Spin%289%29">current</a>

   Added a more recent reference

   • Maurizio Parton, Paolo Piccinni, The Role of Spin(9) in Octonionic Geometry, (arXiv:1810.06288)

   diff, v2, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeDec 26th 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to today's * [[Kirill Krasnov]], _$SO(9)$ characterisation of the Standard Model gauge group_ ([arXiv:1912.11282](https://arxiv.org/abs/1912.11282)) and this text: > The exact [[gauge group]] of the [[standard model of particle physics]] (see [there](standard+model+of+particle+physics#GaugeGroup)) is [[isomorohism|isomorphic]] to the [[subgroup]] of the [[Jordan algebra]] [[automorphism group]] of the [[octonions|octonionic]] [[Albert algebra]] that "[[stabilizer subgroup|stabilizes]] a 4d sub-[[Minkowski spacetime]]" (see [there](Albert+algebra#StabilizerOf4dMinkowskiInsideOctonionicAlbertAlgebra) for details). > More concretely, it is identified with the [[subgroup]] of [[Spin(9)]] which respects a splitting $\mathbb{H} \oplus \mathbb{H} \simeq_{\mathbb{R}} \mathbb{C} \oplus \mathbb{C}^3$ ([Krasnov 19](#Krasnov19)) <a href="https://ncatlab.org/nlab/revision/diff/Spin%289%29/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/Spin%289%29/3">v3</a>, <a href="https://ncatlab.org/nlab/show/Spin%289%29">current</a>

   added pointer to today’s

   • Kirill Krasnov, $SO(9)$ characterisation of the Standard Model gauge group (arXiv:1912.11282)

   and this text:

   The exact gauge group of the standard model of particle physics (see there) is isomorphic to the subgroup of the Jordan algebra automorphism group of the octonionic Albert algebra that “stabilizes a 4d sub-Minkowski spacetime” (see there for details).

   More concretely, it is identified with the subgroup of Spin(9) which respects a splitting $\mathbb{H} \oplus \mathbb{H} \simeq_{\mathbb{R}} \mathbb{C} \oplus \mathbb{C}^3$ (Krasnov 19)

   diff, v3, current

4. • CommentRowNumber4.
   • CommentAuthorDavid_Corfield
   • CommentTimeFeb 26th 2020
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexA couple more references * Thomas Friedrich, _Weak Spin(9)-Structures on 16-dimensional Riemannian Manifolds_, ([arXiv:math/9912112](https://arxiv.org/abs/math/9912112)) * Maurizio Parton, Paolo Piccinni, _Spin(9) and almost complex structures on 16-dimensional manifolds_, ([arXiv:1105.5318](https://arxiv.org/abs/1105.5318)) <a href="https://ncatlab.org/nlab/revision/diff/Spin%289%29/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/Spin%289%29/4">v4</a>, <a href="https://ncatlab.org/nlab/show/Spin%289%29">current</a>

   A couple more references

   • Thomas Friedrich, Weak Spin(9)-Structures on 16-dimensional Riemannian Manifolds, (arXiv:math/9912112)

   • Maurizio Parton, Paolo Piccinni, Spin(9) and almost complex structures on 16-dimensional manifolds, (arXiv:1105.5318)

   diff, v4, current